Cubature methods to solve BSDEs: Error expansion and complexity control

We obtain an explicit error expansion for the solution of Backward Stochastic Differential Equations (BSDEs) using the cubature on Wiener spaces method. The result is proved under a mild strengthening of the assumptions needed for the application of the cubature method. The explicit expansion can then be used to construct implementable higher order approximations via Richardson-Romberg extrapolation. To allow for an effective efficiency improvement of the interpolated algorithm, we introduce an additional projection on finite grids through interpolation operators. We study the resulting complexity reduction in the case of the linear interpolation

A sparse grid approach to balance sheet risk measurement

In this work, we present a numerical method based on a sparse grid approximation to compute the loss distribution of the balance sheet of a financial or an insurance company. We first describe, in a stylised way, the assets and liabilities dynamics that are used for the numerical estimation of the balance sheet distribution. For the pricing and hedging model, we chose a classical Black & Scholes model with a stochastic interest rate following a Hull & White model. The risk management model describing the evolution of the parameters of the pricing and hedging model is a Gaussian model. The new numerical method is compared with the traditional nested simulation approach. We review the convergence of both methods to estimate the risk indicators under consideration. Finally, we provide numerical results showing that the sparse grid approach is extremely competitive for models with moderate dimension

Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems

This paper deals with existence and uniqueness, in viscosity sense, of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. A particular case of this system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem. The switching cost functions are arbitrary. This problem is connected with the valuation of a power plant in the energy market. The main tool is the notion of systems of reflected BSDEs with oblique reflection.Comment: 36 page

Special Libraries, April 1933

Volume 24, Issue 3https://scholarworks.sjsu.edu/sla_sl_1933/1002/thumbnail.jp

A numerical scheme for the quantile hedging problem

We consider numerical approximations to the quantile hedging price of a European claim in a nonlinear market with Markovian dynamics. We study an equivalent stochastic target problem with the conditional probability of success as a new state variable, in addition to the asset value process. We propose numerical approximations based on piecewise constant policy time stepping coupled with novel finite difference schemes. We prove convergence in the monotone case combining backward stochastic differential equation arguments with the Barles and Jakobsen and Barles and Souganidis approaches for nonlinear PDEs. The difficulties compared to the classical setting consist in the construction of monotone schemes under degeneracy due to the perfectly correlated joint process, the unboundedness of the control variable, and the effect of the boundaries in the probability variable on the analysis. We extend the method to a class of nonmonotone schemes using higher order interpolation and prove convergence for linear drivers. In a numerical section, we illustrate the performance of our schemes by considering an example in a financial market with imperfections, and show that a standard nonmonotone scheme produces financially counterintuitive solutions

Chemical surface transformation of SiC-based nanocablesProceedings of the 4th National Conference New Research Trends in Material Science ARM-4 (2005) 2, 596-603.

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laser writing of permanent or updatable micro-inscriptions on photochromic silver nanoparticles-mesoporous TiO2 nanocomposite films

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